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First, in Section 2 we describe a general recipe for getting an nn-plectic manifold from an nn-dimensional field theory. This is already known to people familiar with multisymplectic geometry — but there aren’t many such people. So, it deserves explanation.

An (n+1)(n+1)-form is nondegenerate if the nn-form we get by plugging in one tangent vector is zero only if that vector is zero.
An nn-plectic manifold is a manifold equipped with a closed nondegenerate (n+1)(n+1)-form. When n=1n = 1, an nn-plectic manifold is usually called a symplectic manifold. Symplectic manifolds serve as phase spaces in the classical mechanics of point particles. The path or ‘worldline’ of a point particle is 1-dimensional, so we say n=1n = 1. But the idea generalizes to higher nn!

Suppose we’re studying a field theory where fields are maps
ϕ:Σ→M \phi : \Sigma \to M
where the ‘parameter space’ Σ\Sigma is an nn-dimensional manifold and the ‘target space’ MM is a manifold of any dimension. Then there’s a standard way to build an ‘extended phase space’ for this theory which is an nn-plectic manifold. We describe how this works.

In the case n=1n = 1, a map
ϕ:Σ→M \phi : \Sigma \to M
describes the worldline of a particle moving in the spacetime MM. In this case, the ‘extended phase space’ we’re talking about is just the cotangent bundle T*(Σ×M)T^*(\Sigma \times M). This becomes a symplectic manifold in a well-known way.

We’re mainly interested in the case n=2n = 2, where a map
ϕ:Σ→M \phi : \Sigma \to M
describes the worldsheet of a string moving in the spacetime MM. In this case the extended phase space is a bit more tricky to explain. In our previous draft we just blurted out the answer… but it probably looked quite ad hoc. Now we derive the answer from an already known framework that works for all nn.

Second, in Section 5 we describe how the presence of a BB field affects the 2-plectic structure for a string. This is just like how an electromagnetic field affects the symplectic structure for a particle!

I remember being amazed when I first read that just by modifying the symplectic structure in an obvious way, the equations of motion for a free particle become the equations for a charged particle in an electromagnetic field. No need to change the Hamiltonian! The same thing works for a string in a BB field.

But the main idea of the paper is unchanged: just as a symplectic manifold gives a Lie algebra of observables, a 2-plectic manifold gives a Lie 2-algebra of observables.

It’s well-known that we can describe the dynamics of a particle using its Lie algebra of observables. For example, bracketing with an observable called the Hamiltonian says how other observables change with time. Similarly, we can describe the dynamics of a string using its Lie 2-algebra of observables.

This material is based upon work supported by the National Science Foundation under Grant No. 0653646.

Posted at August 2, 2008 1:56 PM UTC

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Re: Categorified Symplectic Geometry and the Classical String

Thanks for the clue!

I guess Witten is talking about the case where Σ\Sigma is 2-dimensional? That would be about right if you want to get vertex operator algebras.

To understand this, maybe we should drop down a dimension and consider the case where Σ\Sigma is 1-dimensional. Then our extended phase space is T*(Σ×X)T^* (\Sigma \times X). We can imagine trying to do deformation quantization of this phase space and get an algebra. It’s not hard, but suppose you foolishly start with the case where Σ\Sigma and XX are contractible. Then you’re doing deformation quantization to T*(ℝ1×ℝd)T^*(\mathbb{R}^1 \times \mathbb{R}^d), which is easy. You get a Weyl algebra. But if you try to ‘glue together’ these algebras to get a deformation quantization of the whole extended phase space, you’re dealing with a problem which reminds me of ‘sheaves of algebras’. I don’t quite see a sheaf of algebras here, since we only have algebras for contractible open sets — but still, there’s a lot of old work on deformation quantization that starts locally and then uses sheaves, and this reminds me of that.

So, we might imagine something similar is happening one dimension up. We could try to assemble a vertex operator algebra for some interesting target space XX starting from a sheaf of ‘local’ ones. Or something like that.

I’ll have to read the paper a bit more to see if this vague idea is on some right track.

By the way, I think there’s a nice definition of vertex operator algebras using operads. This might allow us to define something analogous to a VOA for other values of nn (the dimension of the parameter space), and then see how to quantize nn-plectic manifolds using sheaves of these ‘nn-algebras’. The case n=1n = 1 should be some familiar stuff about sheaves of Weyl algebras.

Good point, possibly there is a close relation. I wish I had 36 hours a day to go and check that right now.

Just in case it matters for anything: part of the point here is that in 2d there is that famous chiral deRham complex, locally a certain vertex operator algebra, which arranges itself to a sheaf on target space. The crucial point is that there may be non-free 2d CFTs whose non-freedom comes only from global effects. Locally – locally in target space! – they behave like free field theories. More geometrically: if we think of a σ\sigma-model of maps from parameter space to target space and then restrict those maps to land only in contractible open subsets of the target, then all these restrictions are free field theories. Gluing them all together on target space may yield a non-free field theory (or it may fail altogether, if the relevant anomaly=obstruction does not vanish).

Somehow that’s a deep statement, because more naively it would seem that for a sigma-model a local description makes sense only with respect to parameter space. But at least in some cases there is more.

By the way, I think there’s a nice definition of vertex operator algebras using operads.

The vertex operator algebra can be seen to pretty directly encode the properties of functors from the multicategory whose morphisms are Riemann spheres with nn incoming and 1 outgoing puncture to certain graded vector spaces, which are required to depend holomorphically on the insertion points. The crucial ingredient of the VOA, that operation “YY” which eats a complex number zz and a vector uu and spits out a linear map Y(u,z):V→VY(u,z) : V \to V is the image under such a functor of the sphere with three punctures at 00, zz and ∞\infty, respectively.

So the VOA is pretty directly a “holomorphic genus 0” CFT in the sense of Segal, only difference being that Segal has proper circle-boundaries where here one uses parameterized punctures. It’s an imprtant difference but still just a technical one.

Liang Kong taught me a bit about all this here in Bonn. He has a whole series of articles where he takes this observation and runs with it, describing open, closed, oplen-closed CFT at genus 0, genus 1 and potentially higher, relating it to Frobenius algebras and “Cardy algebras” and whatnot in the modular tensor category of reps of the VOA.

Re: Categorified Symplectic Geometry and the Classical String

Re: Categorified Symplectic Geometry and the Classical String

Good point — we should cite Bruce’s work as an example of ‘categorified geometric quantization’. Now the trick will be going from finite groups to compact simple Lie groups, which are blessed with very interesting 2-plectic structures.

Re: Categorified Symplectic Geometry and the Classical String

There Theodore Voronov is using his powerful – by now probably seminal – method of derived brackets to obtain ∞\infty-Poisson algebras. In his setup these come from inhomogeneous multivectorfields which are even graded, and induce on the algebra of functions the structure of an L∞L_\infty-algebra.

I was chatting with Bruce a bit about how this might be related to the Lie 2-algebra on 1-forms which you obtain from 2-plectic structures. Superficially it looks like both constructions are two different kinds of higher versions of ordinary Poisson algebras. But of course they might well be related somehow. Maybe simply by allowing odd multivectorfields in Voronov’s setup. Or maybe it’s more subtle.

Read the post Categorified Symplectic Geometry and the String Lie 2-AlgebraWeblog: The n-Category CaféExcerpt: A new paper shows how to build the string Lie 2-algebra by taking a compact Lie group with its canonical closed 3-form and then using ideas from multisymplectic geometry.Tracked: January 27, 2009 12:03 AM

Re: Categorified Symplectic Geometry and the Classical String

This paper was just accepted for publication in Communications in Mathematical Physics. I’m glad it got accepted now, since Alex Hoffnung is applying for jobs this fall!

Re: Categorified Symplectic Geometry and the Classical String

I might appreciate a helping hand in extracting the list of the more important points from the extensive literature.

I am eventually headed towards discussion of push-forward of differential cocycles along Σ×X→Σ\Sigma \times X \to \Sigma, but for the moment I just want to get the basic traditional theory layed out in front of myself – and any reader who passes by – in a nice nnLab entry.

Re: Categorified Symplectic Geometry and the Classical String

Hi Urs. I’d be happy to help with this. By the way (on a related topic) I’ve been trying out the algorithm you proposed here for obtaining the L∞L_{\infty} algebra of sections from a ∞\infty-Lie algebroid on a few examples, and I am getting some unexpected/strange results. I’d like to discuss it with you at some point (perhaps somewhere in the nlab) if you are still interested in such things.

Re: Categorified Symplectic Geometry and the Classical String

Meanwhile I have added a little bit of genuine substance on “covariant field theory” to the entry. But still pretty incomplete.

Do you know if there is anything at all about geometric quantization on extended phase spaces? I seem to remember some references vaguely related, but now I am not sure.

We should be able to eventualy tell a grand story here: we know that C*C^*-algebraic deformation quantization of a Poisson manifold consists of forming the groupoid algebra of the symplectic Lie groupoid that integrates the corresponding Poisson Lie algebroid.

So we want to eventually head in the direction of understanding the “symplectic Lie nn-groupoids” on extended phase spaces that arise from integrating the corresponding “symplectic/Poisson nn-manifolds”.

It should be very interesting to see the notion of constraints from this perspective of Lie nn-ntegration.

But that will require a bit of thinking. :-)

I’ve been trying out the algorithm you proposed here for obtaining the L ∞ algebra of sections from a ∞-Lie algebroid on a few examples, and I am getting some unexpected/strange results. I’d like to discuss it with you at some point (perhaps somewhere in the nlab) if you are still interested in such things.

Re: Categorified Symplectic Geometry and the Classical String

Kanatchikov has several papers on developing a quantization procedure for multisymplectic geometry. In particular there is this one:

I.V. Kanatchikov, Geometric (pre)quantization in the polysymplectic approach to field theory, arXiv:hep-th/0112263v3.

He also has written a paper on canonical quantization:

DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory, arXiv:hep-th/9810165v1

Kanatchikov’s “algebra of observables” is a Gerstenhaber algebra. The relationship between it and the Lie superalgebra of observables constructed by Forger, Paufler, and Roemer is discussed in this paper: